Ability check

The game rolls an Ability Check to determine whether a creature will succeed or fail in various scenarios. The creature rolls a D20, and adds its Ability Score Modifier and possibly a Proficiency Bonus. To succeed, the result must reach or exceed a Difficulty Class, which depends on what the creature is trying to accomplish.

Result =  d20 + Ability Score Modifier + (optional) Proficiency Bonus

Ability Checks can be based purely on an Ability Score, in which case they can only benefit from the Ability Score Modifier, not the Proficiency Bonus.

Skill Check

More commonly, the check will be related to a specific Skill, which not only determines the Ability Score used for the Modifier, but also lets the creature use its Proficiency Bonus if it happens to be Proficient in that particular Skill. In this case, the check may be called a Skill Check rather than an Ability Check.

For example, when a trap is near a creature, the game rolls a Perception Skill Check. (This can also be called rolling a Perception Check, or simply rolling Perception.) Perception is related to Wisdom, so the creature first of all uses its Wisdom Modifier. If the creature doesn't gain any bonus from Wisdom, it could still gain a bonus from being Proficient in Perception. If the creature both has high Wisdom and is also proficient in Perception, it will gain both bonuses.

Once the trap is seen, the creature may want to disarm it. This will require a Sleight of Hand Check, which is related to Dexterity. So, a creature that both has high Dexterity, and Proficiency in Sleight of Hand, can benefit from both types of bonus when attempting to disarm the trap.

List of Skills, sorted by the Ability Score they're related to

Strength	Dexterity	Intelligence	Wisdom	Charisma
Athletics	Acrobatics Sleight of Hand Stealth	Arcana History Investigation Nature Religion	Animal Handling Insight Medicine Perception Survival	Deception Intimidation Performance Persuasion

(There are no Skills based on Constitution.)

Contest

A special type of Ability Check is called the Contest, in which two creatures both roll an Ability Check or Skill Check to contest each other, and one wins over the other. The creatures don't necessarily roll the same type of check.

An example of this is the Shove action, where the creature attempting the Shove must roll Athletics, and the defending creature is allowed to roll either Athletics or Acrobatics to contest the Shove. If the attacker's roll is higher than the defender's, the Shove succeeds, otherwise it fails.